91Ó°ÊÓ

Understanding the domain and range of a function is crucial in mathematics. The domain of a function includes all the possible input values (x-values) that can be plugged into the function. For the equation \( y = 2x^2 \), the domain is all real numbers because there are no restrictions on the values that x can take. This means x can be anything: negative, zero, or positive.

The range of a function, on the other hand, includes all possible output values (y-values) that result from using the domain values. Since squaring any real number and multiplying by 2 will always yield a non-negative value, the range of the equation \( y = 2x^2 \) is all non-negative real numbers, or \( y \geq 0 \). In simpler terms, y can be any number from zero to positive infinity.

Graphing is a powerful visual tool that helps us understand equations better. To graph the equation \( y=2x^2 \), we start with a set of ordered pairs derived from selected x-values.

• Step 1: Choose values for x, like -2, -1, 0, 1, and 2.
• Step 2: Calculate corresponding y-values by substituting these x-values into the equation. For example, for x = -2, y = 8, making the ordered pair (-2, 8).
• Step 3: List out the ordered pairs: (-2, 8), (-1, 2), (0, 0), (1, 2), (2, 8).
• Step 4: Plot these pairs on a coordinate plane and draw a smooth curve through the points.
  This curve represents the equation graphically.

The graph of \( y=2x^2 \) forms a U-shaped parabola opening upwards, with the vertex at the origin (0,0). Graphing helps easily visualize how y changes with x.

Identifying whether a given relation is a function is essential. A function is a relation where each input (x-value) has exactly one output (y-value). For the equation \( y = 2x^2 \), we need to check if each x-value corresponds to only one y-value.

Let's take a closer look:

• For x = -2, y = 8.
• For x = -1, y = 2.
• For x = 0, y = 0.
• For x = 1, y = 2.
• For x = 2, y = 8.

In each case, every x-value produces a single, unique y-value. Therefore, y is indeed a function of x in the equation \( y = 2x^2 \).

Key takeaway: If you can define a unique y for every x in an equation, you've identified a function. This understanding is fundamental in higher mathematics and calculus, where the concept of functions is widely used.